Complete Cayley Tables and Every Finite Group Obligation
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261013-advance-20261022-r05-finite-group-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
2. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
3. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[1, 3, 0, 4, 2], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [4, 2, 3, 0, 1], [2, 0, 4, 1, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[1, 3, 0, 4, 2], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [4, 2, 3, 0, 1], [2, 0, 4, 1, 3]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[1, 3, 0, 4, 2], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [4, 2, 3, 0, 1], [2, 0, 4, 1, 3]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
4. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [0, 1, 2, 3, 4], [3, 0, 1, 4, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [0, 1, 2, 3, 4], [3, 0, 1, 4, 2]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [0, 1, 2, 3, 4], [3, 0, 1, 4, 2]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
5. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
6. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
7. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[4, 0, 3, 1, 2], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[4, 0, 3, 1, 2], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 0, 3, 1, 2], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
8. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [2, 4, 3, 1, 0], [1, 2, 4, 0, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [2, 4, 3, 1, 0], [1, 2, 4, 0, 3]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [2, 4, 3, 1, 0], [1, 2, 4, 0, 3]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
9. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
10. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
11. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
12. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
13. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [3, 0, 4, 2, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [3, 0, 4, 2, 1]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [3, 0, 4, 2, 1]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
14. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
15. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
16. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
17. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [4, 3, 0, 2, 1], [2, 4, 3, 1, 0]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [4, 3, 0, 2, 1], [2, 4, 3, 1, 0]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [4, 3, 0, 2, 1], [2, 4, 3, 1, 0]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
18. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[3, 2, 0, 4, 1], [2, 4, 1, 0, 3], [0, 1, 2, 3, 4], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[3, 2, 0, 4, 1], [2, 4, 1, 0, 3], [0, 1, 2, 3, 4], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 2, 0, 4, 1], [2, 4, 1, 0, 3], [0, 1, 2, 3, 4], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
19. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
20. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [1, 2, 4, 0, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [1, 2, 4, 0, 3], [0, 1, 2, 3, 4]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [1, 2, 4, 0, 3], [0, 1, 2, 3, 4]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
21. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
22. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[3, 0, 4, 2, 1], [0, 1, 2, 3, 4], [4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [1, 4, 3, 0, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[3, 0, 4, 2, 1], [0, 1, 2, 3, 4], [4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [1, 4, 3, 0, 2]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 0, 4, 2, 1], [0, 1, 2, 3, 4], [4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [1, 4, 3, 0, 2]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
23. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
24. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[2, 4, 1, 0, 3], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[2, 4, 1, 0, 3], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 4, 1, 0, 3], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
25. Closure of Every Cell in a Complete Cayley Table
The complete finite Cayley table=[[2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [1, 4, 3, 0, 2], [4, 2, 0, 1, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. This family permits out-of-carrier entries to test qualification. Give (carrier size,table,does every entry lie in the carrier?).
View answer and explanation
(5, [[2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [1, 4, 3, 0, 2], [4, 2, 0, 1, 3], [0, 1, 2, 3, 4]], Yes)
Inspect every table cell. Closure alone does not imply associativity, an identity or inverses. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [1, 4, 3, 0, 2], [4, 2, 0, 1, 3], [0, 1, 2, 3, 4]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-table-qualification · candidate
26. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
27. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
28. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[1, 3, 0, 4, 2], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [4, 2, 3, 0, 1], [2, 0, 4, 1, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[1, 3, 0, 4, 2], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [4, 2, 3, 0, 1], [2, 0, 4, 1, 3]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[1, 3, 0, 4, 2], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [4, 2, 3, 0, 1], [2, 0, 4, 1, 3]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
29. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [0, 1, 2, 3, 4], [3, 0, 1, 4, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [0, 1, 2, 3, 4], [3, 0, 1, 4, 2]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [0, 1, 2, 3, 4], [3, 0, 1, 4, 2]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
30. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
31. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
32. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[4, 0, 3, 1, 2], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[4, 0, 3, 1, 2], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 0, 3, 1, 2], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
33. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [2, 4, 3, 1, 0], [1, 2, 4, 0, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [2, 4, 3, 1, 0], [1, 2, 4, 0, 3]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [2, 4, 3, 1, 0], [1, 2, 4, 0, 3]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
34. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
35. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
36. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
37. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
38. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [3, 0, 4, 2, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [3, 0, 4, 2, 1]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [3, 0, 4, 2, 1]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
39. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
40. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
41. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
42. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [4, 3, 0, 2, 1], [2, 4, 3, 1, 0]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [4, 3, 0, 2, 1], [2, 4, 3, 1, 0]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [4, 3, 0, 2, 1], [2, 4, 3, 1, 0]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
43. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[3, 2, 0, 4, 1], [2, 4, 1, 0, 3], [0, 1, 2, 3, 4], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[3, 2, 0, 4, 1], [2, 4, 1, 0, 3], [0, 1, 2, 3, 4], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 2, 0, 4, 1], [2, 4, 1, 0, 3], [0, 1, 2, 3, 4], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
44. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
45. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [1, 2, 4, 0, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [1, 2, 4, 0, 3], [0, 1, 2, 3, 4]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [1, 2, 4, 0, 3], [0, 1, 2, 3, 4]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
46. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
47. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[3, 0, 4, 2, 1], [0, 1, 2, 3, 4], [4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [1, 4, 3, 0, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[3, 0, 4, 2, 1], [0, 1, 2, 3, 4], [4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [1, 4, 3, 0, 2]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 0, 4, 2, 1], [0, 1, 2, 3, 4], [4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [1, 4, 3, 0, 2]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
48. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
49. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[2, 4, 1, 0, 3], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[2, 4, 1, 0, 3], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 4, 1, 0, 3], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
50. Exhaustive Associativity over All Ordered Triples
The complete finite Cayley table=[[2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [1, 4, 3, 0, 2], [4, 2, 0, 1, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require every entry in the carrier. Give (size,table,every [a,b,c] violating (a*b)*c=a*(b*c),associative?). List bad triples in nested increasing a,b,c order, including all triples with repeated elements.
View answer and explanation
(5, [[2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [1, 4, 3, 0, 2], [4, 2, 0, 1, 3], [0, 1, 2, 3, 4]], [], Yes)
Compare both parenthesizations for every ordered triple. Symmetry or a few successful cases does not replace this exhaustive check. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [1, 4, 3, 0, 2], [4, 2, 0, 1, 3], [0, 1, 2, 3, 4]], [], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-associativity-audit · candidate
51. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], [0], [[0, [0]], [1, [4]], [2, [3]], [3, [2]], [4, [1]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], [0], [[0, [0]], [1, [4]], [2, [3]], [3, [2]], [4, [1]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
52. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]], [1], [[0, [4]], [1, [1]], [2, [3]], [3, [2]], [4, [0]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]], [1], [[0, [4]], [1, [1]], [2, [3]], [3, [2]], [4, [0]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
53. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[1, 3, 0, 4, 2], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [4, 2, 3, 0, 1], [2, 0, 4, 1, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[1, 3, 0, 4, 2], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [4, 2, 3, 0, 1], [2, 0, 4, 1, 3]], [2], [[0, [4]], [1, [3]], [2, [2]], [3, [1]], [4, [0]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[1, 3, 0, 4, 2], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [4, 2, 3, 0, 1], [2, 0, 4, 1, 3]], [2], [[0, [4]], [1, [3]], [2, [2]], [3, [1]], [4, [0]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
54. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [0, 1, 2, 3, 4], [3, 0, 1, 4, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [0, 1, 2, 3, 4], [3, 0, 1, 4, 2]], [3], [[0, [4]], [1, [2]], [2, [1]], [3, [3]], [4, [0]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [0, 1, 2, 3, 4], [3, 0, 1, 4, 2]], [3], [[0, [4]], [1, [2]], [2, [1]], [3, [3]], [4, [0]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
55. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4]], [4], [[0, [3]], [1, [2]], [2, [1]], [3, [0]], [4, [4]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4]], [4], [[0, [3]], [1, [2]], [2, [1]], [3, [0]], [4, [4]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
56. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]], [0], [[0, [0]], [1, [3]], [2, [4]], [3, [1]], [4, [2]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]], [0], [[0, [0]], [1, [3]], [2, [4]], [3, [1]], [4, [2]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
57. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[4, 0, 3, 1, 2], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[4, 0, 3, 1, 2], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3]], [1], [[0, [3]], [1, [1]], [2, [4]], [3, [0]], [4, [2]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 0, 3, 1, 2], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3]], [1], [[0, [3]], [1, [1]], [2, [4]], [3, [0]], [4, [2]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
58. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [2, 4, 3, 1, 0], [1, 2, 4, 0, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [2, 4, 3, 1, 0], [1, 2, 4, 0, 3]], [2], [[0, [3]], [1, [4]], [2, [2]], [3, [0]], [4, [1]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [2, 4, 3, 1, 0], [1, 2, 4, 0, 3]], [2], [[0, [3]], [1, [4]], [2, [2]], [3, [0]], [4, [1]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
59. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], [3], [[0, [2]], [1, [4]], [2, [0]], [3, [3]], [4, [1]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], [3], [[0, [2]], [1, [4]], [2, [0]], [3, [3]], [4, [1]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
60. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], [4], [[0, [2]], [1, [3]], [2, [0]], [3, [1]], [4, [4]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], [4], [[0, [2]], [1, [3]], [2, [0]], [3, [1]], [4, [4]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
61. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]], [0], [[0, [0]], [1, [2]], [2, [1]], [3, [4]], [4, [3]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]], [0], [[0, [0]], [1, [2]], [2, [1]], [3, [4]], [4, [3]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
62. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2]], [1], [[0, [2]], [1, [1]], [2, [0]], [3, [4]], [4, [3]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2]], [1], [[0, [2]], [1, [1]], [2, [0]], [3, [4]], [4, [3]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
63. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [3, 0, 4, 2, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [3, 0, 4, 2, 1]], [2], [[0, [1]], [1, [0]], [2, [2]], [3, [4]], [4, [3]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [3, 0, 4, 2, 1]], [2], [[0, [1]], [1, [0]], [2, [2]], [3, [4]], [4, [3]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
64. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], [3], [[0, [1]], [1, [0]], [2, [4]], [3, [3]], [4, [2]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], [3], [[0, [1]], [1, [0]], [2, [4]], [3, [3]], [4, [2]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
65. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], [4], [[0, [1]], [1, [0]], [2, [3]], [3, [2]], [4, [4]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], [4], [[0, [1]], [1, [0]], [2, [3]], [3, [2]], [4, [4]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
66. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]], [0], [[0, [0]], [1, [2]], [2, [1]], [3, [4]], [4, [3]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]], [0], [[0, [0]], [1, [2]], [2, [1]], [3, [4]], [4, [3]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
67. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [4, 3, 0, 2, 1], [2, 4, 3, 1, 0]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [4, 3, 0, 2, 1], [2, 4, 3, 1, 0]], [1], [[0, [2]], [1, [1]], [2, [0]], [3, [4]], [4, [3]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [4, 3, 0, 2, 1], [2, 4, 3, 1, 0]], [1], [[0, [2]], [1, [1]], [2, [0]], [3, [4]], [4, [3]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
68. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[3, 2, 0, 4, 1], [2, 4, 1, 0, 3], [0, 1, 2, 3, 4], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[3, 2, 0, 4, 1], [2, 4, 1, 0, 3], [0, 1, 2, 3, 4], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0]], [2], [[0, [1]], [1, [0]], [2, [2]], [3, [4]], [4, [3]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 2, 0, 4, 1], [2, 4, 1, 0, 3], [0, 1, 2, 3, 4], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0]], [2], [[0, [1]], [1, [0]], [2, [2]], [3, [4]], [4, [3]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
69. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0]], [3], [[0, [1]], [1, [0]], [2, [4]], [3, [3]], [4, [2]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0]], [3], [[0, [1]], [1, [0]], [2, [4]], [3, [3]], [4, [2]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
70. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [1, 2, 4, 0, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [1, 2, 4, 0, 3], [0, 1, 2, 3, 4]], [4], [[0, [1]], [1, [0]], [2, [3]], [3, [2]], [4, [4]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [1, 2, 4, 0, 3], [0, 1, 2, 3, 4]], [4], [[0, [1]], [1, [0]], [2, [3]], [3, [2]], [4, [4]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
71. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]], [0], [[0, [0]], [1, [4]], [2, [3]], [3, [2]], [4, [1]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]], [0], [[0, [0]], [1, [4]], [2, [3]], [3, [2]], [4, [1]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
72. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[3, 0, 4, 2, 1], [0, 1, 2, 3, 4], [4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [1, 4, 3, 0, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[3, 0, 4, 2, 1], [0, 1, 2, 3, 4], [4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [1, 4, 3, 0, 2]], [1], [[0, [4]], [1, [1]], [2, [3]], [3, [2]], [4, [0]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 0, 4, 2, 1], [0, 1, 2, 3, 4], [4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [1, 4, 3, 0, 2]], [1], [[0, [4]], [1, [1]], [2, [3]], [3, [2]], [4, [0]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
73. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], [2], [[0, [4]], [1, [3]], [2, [2]], [3, [1]], [4, [0]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], [2], [[0, [4]], [1, [3]], [2, [2]], [3, [1]], [4, [0]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
74. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[2, 4, 1, 0, 3], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[2, 4, 1, 0, 3], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1]], [3], [[0, [4]], [1, [2]], [2, [1]], [3, [3]], [4, [0]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 4, 1, 0, 3], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1]], [3], [[0, [4]], [1, [2]], [2, [1]], [3, [3]], [4, [0]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
75. Two-Sided Identity and Two-Sided Inverse Inventories
The complete finite Cayley table=[[2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [1, 4, 3, 0, 2], [4, 2, 0, 1, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require closure. Give (size,table,all two-sided identity labels,every [x,all two-sided inverse labels] row relative to a unique identity,all elements have such an inverse relative to a unique identity?). If no unique identity is available, output an empty inverse-row list and a final No.
View answer and explanation
(5, [[2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [1, 4, 3, 0, 2], [4, 2, 0, 1, 3], [0, 1, 2, 3, 4]], [4], [[0, [3]], [1, [2]], [2, [1]], [3, [0]], [4, [4]]], Yes)
Require both e*x=x and x*e=x. Inverse candidates must satisfy both x*y=e and y*x=e for the same identity. This inventory alone does not audit associativity. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [1, 4, 3, 0, 2], [4, 2, 0, 1, 3], [0, 1, 2, 3, 4]], [4], [[0, [3]], [1, [2]], [2, [1]], [3, [0]], [4, [4]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-identity-inverse-inventory · candidate
76. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], Yes, Yes, [0], [[0, [0]], [1, [4]], [2, [3]], [3, [2]], [4, [1]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3]], Yes, Yes, [0], [[0, [0]], [1, [4]], [2, [3]], [3, [2]], [4, [1]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
77. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]], Yes, Yes, [1], [[0, [4]], [1, [1]], [2, [3]], [3, [2]], [4, [0]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 0, 3, 4, 1], [0, 1, 2, 3, 4], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3]], Yes, Yes, [1], [[0, [4]], [1, [1]], [2, [3]], [3, [2]], [4, [0]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
78. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[1, 3, 0, 4, 2], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [4, 2, 3, 0, 1], [2, 0, 4, 1, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[1, 3, 0, 4, 2], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [4, 2, 3, 0, 1], [2, 0, 4, 1, 3]], Yes, Yes, [2], [[0, [4]], [1, [3]], [2, [2]], [3, [1]], [4, [0]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[1, 3, 0, 4, 2], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [4, 2, 3, 0, 1], [2, 0, 4, 1, 3]], Yes, Yes, [2], [[0, [4]], [1, [3]], [2, [2]], [3, [1]], [4, [0]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
79. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [0, 1, 2, 3, 4], [3, 0, 1, 4, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [0, 1, 2, 3, 4], [3, 0, 1, 4, 2]], Yes, Yes, [3], [[0, [4]], [1, [2]], [2, [1]], [3, [3]], [4, [0]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[1, 2, 4, 0, 3], [2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [0, 1, 2, 3, 4], [3, 0, 1, 4, 2]], Yes, Yes, [3], [[0, [4]], [1, [2]], [2, [1]], [3, [3]], [4, [0]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
80. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4]], Yes, Yes, [4], [[0, [3]], [1, [2]], [2, [1]], [3, [0]], [4, [4]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4]], Yes, Yes, [4], [[0, [3]], [1, [2]], [2, [1]], [3, [0]], [4, [4]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
81. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]], Yes, Yes, [0], [[0, [0]], [1, [3]], [2, [4]], [3, [1]], [4, [2]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [4, 2, 0, 1, 3]], Yes, Yes, [0], [[0, [0]], [1, [3]], [2, [4]], [3, [1]], [4, [2]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
82. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[4, 0, 3, 1, 2], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[4, 0, 3, 1, 2], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3]], Yes, Yes, [1], [[0, [3]], [1, [1]], [2, [4]], [3, [0]], [4, [2]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 0, 3, 1, 2], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3]], Yes, Yes, [1], [[0, [3]], [1, [1]], [2, [4]], [3, [0]], [4, [2]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
83. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [2, 4, 3, 1, 0], [1, 2, 4, 0, 3]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [2, 4, 3, 1, 0], [1, 2, 4, 0, 3]], Yes, Yes, [2], [[0, [3]], [1, [4]], [2, [2]], [3, [0]], [4, [1]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [2, 4, 3, 1, 0], [1, 2, 4, 0, 3]], Yes, Yes, [2], [[0, [3]], [1, [4]], [2, [2]], [3, [0]], [4, [1]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
84. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], Yes, Yes, [3], [[0, [2]], [1, [4]], [2, [0]], [3, [3]], [4, [1]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 2, 3, 0, 1], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [0, 1, 2, 3, 4], [1, 3, 0, 4, 2]], Yes, Yes, [3], [[0, [2]], [1, [4]], [2, [0]], [3, [3]], [4, [1]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
85. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], Yes, Yes, [4], [[0, [2]], [1, [3]], [2, [0]], [3, [1]], [4, [4]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 2, 4, 1, 0], [2, 0, 3, 4, 1], [4, 3, 1, 0, 2], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4]], Yes, Yes, [4], [[0, [2]], [1, [3]], [2, [0]], [3, [1]], [4, [4]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
86. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]], Yes, Yes, [0], [[0, [0]], [1, [2]], [2, [1]], [3, [4]], [4, [3]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 4, 0, 2, 3], [2, 0, 3, 4, 1], [3, 2, 4, 1, 0], [4, 3, 1, 0, 2]], Yes, Yes, [0], [[0, [0]], [1, [2]], [2, [1]], [3, [4]], [4, [3]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
87. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2]], Yes, Yes, [1], [[0, [2]], [1, [1]], [2, [0]], [3, [4]], [4, [3]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1], [3, 4, 0, 1, 2]], Yes, Yes, [1], [[0, [2]], [1, [1]], [2, [0]], [3, [4]], [4, [3]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
88. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [3, 0, 4, 2, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [3, 0, 4, 2, 1]], Yes, Yes, [2], [[0, [1]], [1, [0]], [2, [2]], [3, [4]], [4, [3]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [0, 1, 2, 3, 4], [1, 4, 3, 0, 2], [3, 0, 4, 2, 1]], Yes, Yes, [2], [[0, [1]], [1, [0]], [2, [2]], [3, [4]], [4, [3]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
89. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], Yes, Yes, [3], [[0, [1]], [1, [0]], [2, [4]], [3, [3]], [4, [2]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[4, 3, 1, 0, 2], [3, 2, 4, 1, 0], [1, 4, 0, 2, 3], [0, 1, 2, 3, 4], [2, 0, 3, 4, 1]], Yes, Yes, [3], [[0, [1]], [1, [0]], [2, [4]], [3, [3]], [4, [2]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
90. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], Yes, Yes, [4], [[0, [1]], [1, [0]], [2, [3]], [3, [2]], [4, [4]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 4, 1, 2, 0], [4, 2, 3, 0, 1], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [0, 1, 2, 3, 4]], Yes, Yes, [4], [[0, [1]], [1, [0]], [2, [3]], [3, [2]], [4, [4]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
91. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]], Yes, Yes, [0], [[0, [0]], [1, [2]], [2, [1]], [3, [4]], [4, [3]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 3, 0, 4, 2], [2, 0, 4, 1, 3], [3, 4, 1, 2, 0], [4, 2, 3, 0, 1]], Yes, Yes, [0], [[0, [0]], [1, [2]], [2, [1]], [3, [4]], [4, [3]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
92. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [4, 3, 0, 2, 1], [2, 4, 3, 1, 0]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [4, 3, 0, 2, 1], [2, 4, 3, 1, 0]], Yes, Yes, [1], [[0, [2]], [1, [1]], [2, [0]], [3, [4]], [4, [3]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 0, 1, 4, 2], [0, 1, 2, 3, 4], [1, 2, 4, 0, 3], [4, 3, 0, 2, 1], [2, 4, 3, 1, 0]], Yes, Yes, [1], [[0, [2]], [1, [1]], [2, [0]], [3, [4]], [4, [3]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
93. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[3, 2, 0, 4, 1], [2, 4, 1, 0, 3], [0, 1, 2, 3, 4], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[3, 2, 0, 4, 1], [2, 4, 1, 0, 3], [0, 1, 2, 3, 4], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0]], Yes, Yes, [2], [[0, [1]], [1, [0]], [2, [2]], [3, [4]], [4, [3]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 2, 0, 4, 1], [2, 4, 1, 0, 3], [0, 1, 2, 3, 4], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0]], Yes, Yes, [2], [[0, [1]], [1, [0]], [2, [2]], [3, [4]], [4, [3]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
94. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0]], Yes, Yes, [3], [[0, [1]], [1, [0]], [2, [4]], [3, [3]], [4, [2]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 3, 4, 0, 1], [3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0]], Yes, Yes, [3], [[0, [1]], [1, [0]], [2, [4]], [3, [3]], [4, [2]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
95. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [1, 2, 4, 0, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [1, 2, 4, 0, 3], [0, 1, 2, 3, 4]], Yes, Yes, [4], [[0, [1]], [1, [0]], [2, [3]], [3, [2]], [4, [4]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 4, 3, 1, 0], [4, 3, 0, 2, 1], [3, 0, 1, 4, 2], [1, 2, 4, 0, 3], [0, 1, 2, 3, 4]], Yes, Yes, [4], [[0, [1]], [1, [0]], [2, [3]], [3, [2]], [4, [4]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
96. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]], Yes, Yes, [0], [[0, [0]], [1, [4]], [2, [3]], [3, [2]], [4, [1]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[0, 1, 2, 3, 4], [1, 3, 4, 2, 0], [2, 4, 1, 0, 3], [3, 2, 0, 4, 1], [4, 0, 3, 1, 2]], Yes, Yes, [0], [[0, [0]], [1, [4]], [2, [3]], [3, [2]], [4, [1]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
97. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[3, 0, 4, 2, 1], [0, 1, 2, 3, 4], [4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [1, 4, 3, 0, 2]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[3, 0, 4, 2, 1], [0, 1, 2, 3, 4], [4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [1, 4, 3, 0, 2]], Yes, Yes, [1], [[0, [4]], [1, [1]], [2, [3]], [3, [2]], [4, [0]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 0, 4, 2, 1], [0, 1, 2, 3, 4], [4, 2, 0, 1, 3], [2, 3, 1, 4, 0], [1, 4, 3, 0, 2]], Yes, Yes, [1], [[0, [4]], [1, [1]], [2, [3]], [3, [2]], [4, [0]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
98. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], Yes, Yes, [2], [[0, [4]], [1, [3]], [2, [2]], [3, [1]], [4, [0]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[3, 4, 0, 1, 2], [4, 0, 1, 2, 3], [0, 1, 2, 3, 4], [1, 2, 3, 4, 0], [2, 3, 4, 0, 1]], Yes, Yes, [2], [[0, [4]], [1, [3]], [2, [2]], [3, [1]], [4, [0]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
99. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[2, 4, 1, 0, 3], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[2, 4, 1, 0, 3], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1]], Yes, Yes, [3], [[0, [4]], [1, [2]], [2, [1]], [3, [3]], [4, [0]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 4, 1, 0, 3], [4, 0, 3, 1, 2], [1, 3, 4, 2, 0], [0, 1, 2, 3, 4], [3, 2, 0, 4, 1]], Yes, Yes, [3], [[0, [4]], [1, [2]], [2, [1]], [3, [3]], [4, [0]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
100. Close Every Group Obligation on the Specified Finite Carrier
The complete finite Cayley table=[[2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [1, 4, 3, 0, 2], [4, 2, 0, 1, 3], [0, 1, 2, 3, 4]] uses elements Fin(5)={0,…,4} and entry table[a][b]=a*b. Preserve element labels and row/column order. A sampled row or triple does not prove the group axioms. Require table entries in the carrier. Give (size,table,closed?,associative?,two-sided identities,inverse rows relative to a unique identity,is this table a group?). Audit all ordered triples and all elements, not samples.
View answer and explanation
(5, [[2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [1, 4, 3, 0, 2], [4, 2, 0, 1, 3], [0, 1, 2, 3, 4]], Yes, Yes, [4], [[0, [3]], [1, [2]], [2, [1]], [3, [0]], [4, [4]]], Yes)
A group requires closure, complete associativity, a two-sided identity and two-sided inverses for every element. Preserve the labels and all inverse candidates. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is (5, [[2, 3, 1, 4, 0], [3, 0, 4, 2, 1], [1, 4, 3, 0, 2], [4, 2, 0, 1, 3], [0, 1, 2, 3, 4]], Yes, Yes, [4], [[0, [3]], [1, [2]], [2, [1]], [3, [0]], [4, [4]]], Yes).
Reviewed step-by-step derivation
Finite group · qf-str22-group-full-group-audit · candidate
Canonical question JSONL · English JSONL · Validation report